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Quantitative bounds for large deviations of heavy tailed random variables

Quirin Vogel

Abstract

The probability that the sum of independent, centered, identically distributed, heavy-tailed random variables achieves a very large value is asymptotically equal to the probability that there exists a single summand equalling that value. We quantify the error in this approximation. We furthermore characterise of the law of the individual summands, conditioned on the sum being large.

Quantitative bounds for large deviations of heavy tailed random variables

Abstract

The probability that the sum of independent, centered, identically distributed, heavy-tailed random variables achieves a very large value is asymptotically equal to the probability that there exists a single summand equalling that value. We quantify the error in this approximation. We furthermore characterise of the law of the individual summands, conditioned on the sum being large.
Paper Structure (8 sections, 6 theorems, 105 equations)

This paper contains 8 sections, 6 theorems, 105 equations.

Table of Contents

  1. Introduction and setting
  2. Results
  3. An application
  4. Proofs
  5. Technical preliminaries
  6. Proof of Theorem \ref{['Thm1']}
  7. Proof of Theorem \ref{['ThmNonLocal']}
  8. Proof of Theorem \ref{['Thm2']} and Theorem \ref{['Thm3']}

Key Result

Theorem 2.1

Suppose that $L$ is slowly varying with precision $\mathrm{err}[x,y]$. Assume Equation RightTail holds with $p>0$ and $\mathbb{P}(X_1<-x)\le {\mathcal{O} }(1) L(x)x^{-\tilde{\alpha}}$ holds with some $\tilde{\alpha}\ge \alpha$ (as $x\to \infty$). Set $\widehat{S}_n=S_n-\left\lfloor b_n \right\rfloor We then have that for every $\varepsilon>0$ small enough See Remark RemarkGaussianCase for the sli

Theorems & Definitions (13)

  • Theorem 2.1
  • Example 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Example 2.6
  • Theorem 2.7
  • Proposition 3.1
  • proof
  • Remark 4.1: The Gaussian domain of attraction
  • ...and 3 more